We employ advanced physical and mathematical methods to understand the biophysics of complex cellular processes. A major emphasis has been on the biogenesis of coated vesicles involved in clathrin mediated endocytosis (CME) and other intracellular transport processes. CME is the principal pathway for the regulation of receptors and internalization of certain nutrients and signaling molecules at the plasma membrane of eukaryotic cells. Defects in CME can lead to metabolic disorders, aberrant signaling related to various cancers, and neurological disorders. The process is central to several emerging biotechnological processes, including targeted drug delivery and gene transfer. The early stage of receptor mediated endocytosis involves the formation of transient structures known as clathrin coated pits (CCPs) which, depending on the detailed energetics of protein binding and associated membrane transformations, either mature into clathrin coated vesicles (CCVs) or regress and vanish from the cell surface. The former are referred to as productive CCPs and the latter as abortive CCPs. We have developed a simple physical model for CCP dynamics and have carried out Monte Carlo simulations to investigate the time development of CCP size and explain the origin of abortive pits and features of their lifetime distribution. By fitting the results of the simulations to experimental data, we have been able to estimate values of the free energy changes involved in formation of the clathrin-associated protein complexes that comprise the coat, and have shown how the binding of cargo might modify the coat parameters and thereby facilitate CCV formation. We also have derived analytical expressions for the lifetime distribution and the distribution of maximum sizes of abortive pits, which may be useful in extracting additional information about the mechanism of CCP assembly from experimental data. Moreover, we have obtained a mathematical expression for the stochastic fate of a nascent pit, i.e., whether it will disassemble or mature into clathrin coated vesicles. This generalized expression is being used to identify parameters which affect processes in which clathrin-mediated endocytosis plays a role. In particular, we are investigating nanoparticles that are employed as drug delivery vehicles, in order to establish criteria that might be used when optimizing their design. We also are investigating the role that RME might play in the formation of morphogen gradients and resultant patterning in developing tissues. Part of our research also involves adapting physical methods such as dynamical light scattering, small angle neutron scattering, and atomic force spectroscopy to infer the physical properties of the biochemical entities involved in RME. Due to the unusual properties of clathrin (particularly in the form of the three-legged supramolecular unit called a 'triskelion'), our analysis usually requires the derivation of new mathematical theories and computational algorithms to link experimental observables to underlying biomolecular structure. For example, recent work has used atomic force microscopy (AFM) and single molecule force spectroscopy (SMFS) to characterize intermolecular interactions and domains of clathrin triskelions and assemblies thereof. To assess triskelion structure and triskelion-triskelion interactions, we subjected purified individual triskelions, bovine-brain CCVs, and reconstituted clathrin-AP180 coats to AFM-SMFS pulling experiments and applied newly-derived analytics to extract force-extension relations from very large data sets. For individual triskelions, SMFS reveals a series of unfolding events associated with individual heavy chain alpha-helix hairpins containing ca. 30 amino acid residues. Cooperative unraveling of several hairpin domains up to the size of the known repeating motif of ca. 145 amino acid residues is also seen. We find that the clathrin lattices of AP180-mediated coats are energetically easier to unravel than those of native CCVs. Studies of such clathrin assemblies expose weaker, but coordinated, clathrin-clathrin interactions that are indicative of the inter-leg associations essential for clathrin mediated endocytosis. This investigation is a continuation of our earlier work to establish various mechanical characteristics of clathrin structures, such properties being important elements in physical models such as those mentioned above. In a subsidiary project involving mechanical aspects of cell response, we established a reliable method to assess the coupling between substrate properties and fundamental cell processes such as angiogenesis, neurogenesis and cancer metastasis, all of which are thought to be modulated by extracellular matrix stiffness. The availability of matrix substrates having well-defined stiffness profiles can be of great importance in biophysical studies of cell-substrate interaction. We thus developed a method to fabricate bio-compatible hydrogels having well defined and linear stiffness gradients. This method, involving the photopolymerization of films by progressively uncovering an acrylamide/bis-acrylamide solution initially covered with an opaque mask, can be easily implemented with common lab equipment. It produces linear stiffness gradients of at least 40 kPa/mm, extending from <1 kPa to 80 kPa (in units of shear modulus). These hydrogels were covalently functionalized with uniform coatings of proteins that promote cell adhesion. We found that cell spreading linearly correlates with hydrogel stiffness, indicating that this technique effectively modifies the mechanical environment to which living cells are sensitive. An extension of this work might focus on the collective movements of mechanically-interacting cells, with applications in studies of wound healing, cancer metastasis, and normal and aberrant development of embryonic tissues.